Svetlana Zhilina

• PhD
• Associate Professor at Lomonosov Moscow State University

For the ring of square matrices Matn(𝕜) of order n over a field 𝕜, one can construct the orthogonality graph O(Matn(𝕜)), whose vertices are the zero divisors of the ring Matn(𝕜). Two vertices A and B are connected by an edge if AB = BA = 0. The notion of the distance between two elements of the ring naturally implies that one can consider the set \...

We aim to classify Hilbert \(C^*\)-modules based on nonlinear structure of strong Birkhoff–James orthogonality. We do this by studying purely graph-theoretical properties of a digraph induced by the relation of strong Birkhoff–James orthogonality.

The commutativity graph of the real sedenion algebra is considered. It is shown that the elements whose imaginary parts are not zero divisors correspond to isolated vertices of this graph. All other elements form a connected component whose diameter equals 3.

For an arbitrary normed space X over a field F ∈ {R, C}, we define the directed graph Γ(X) induced by Birkhoff-James orthogonality on the projective space P(X), and also its nonprojective counterpart Γ_0(X). We show that, in finite-dimensional normed spaces, Γ(X) carries all the information about the dimension, smooth points, and norm's maximal fac...

Zero divisors of Cayley–Dickson algebras over an arbitrary field 𝔽, char 𝔽 ≠= 2, are studied. It is shown that the zero divisors whose components alternate strongly pairwise and have nonzero norm form hexagonal structures in the zero-divisor graph of a Cayley–Dickson algebra. Properties of the doubly alternative zero divisors at least one of whose...

Our paper is devoted to the investigations of doubly alternative zero divisors of the real Cayley–Dickson split-algebras. We describe their annihilators and orthogonalizers and also establish the relationship between centralizers and orthogonalizers for such elements. Then we obtain an analogue of the real Jordan normal form in the case of the spli...

The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial Cf(x). We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Di...

We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial $f(x)...

We present Birkhoff–James orthogonality from historical perspectives to the current development. We compare it with some other orthogonalities, present its properties and its applications, and review the characterizations of Birkhoff–James orthogonality in classical Banach spaces like \(\mathbb B(\mathcal {H})\), C ∗-algebras, Hilbert C ∗-modules,...

We suggest a new method which allows us to compute the lengths of (possibly non-unital) standard composition algebras over an arbitrary field F with char F≠2.

It is shown that Birkhoff-James orthogonality knows everything about the smooth norms in reflexive Banach spaces and can also compute the dimensions of the underlying normed spaces.

Let 𝕊 denote the algebra of sedenions and let ΓO(𝕊) denote its orthogonality graph. One can observe that every pair of zero divisors in 𝕊 generates a double hexagon in ΓO(𝕊). The set of vertices of a double hexagon can be extended to a basis of 𝕊 that has a convenient multiplication table. The set of vertices of an arbitrary connected component of...

We consider zero divisors of an arbitrary real Cayley–Dickson algebra such that their components are both standard basis elements. We construct inductively the orthogonality graph on these elements. Then we show that, if we restrict our attention to at least [Formula: see text]-dimensional algebras, two algebras are isomorphic if and only if their...

We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the...

Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise...

We classify surjective linear maps on B(H) that preserve mutual strong Birkhoff–James orthogonality.

The paper introduces the Cayley–Dickson split-sedenion algebra. Exact expressions for the annihilators and orthogonalizers of its zero divisors are obtained, and these results are applied in describing relation graphs of the split-sedenions in terms of their diameters and cliques.

We study the relation of mutual strong Birkhoff–James orthogonality in two classical \(C^*\)-algebras: the \(C^*\)-algebra \({\mathbb {B}}(H)\) of all bounded linear operators on a complex Hilbert space H and the commutative, possibly nonunital, \(C^*\)-algebra. With the help of the induced graph it is shown that this relation alone can characteriz...

The paper studies the anticommutativity condition for elements of arbitrary real Cayley–Dickson algebras. As a consequence, the anticommutativity graphs on equivalence classes of such algebras are classified. Under some additional assumptions on the algebras considered, an expression for the centralizer of an element in terms of its orthogonalizer...